Problem: Find the sum of the first $8$ terms in the following geometric series. Do not round your answer. $2+8+32+...$
Explanation: This formula gives the sum ${S_n}$ of the first $ n$ terms in the geometric series where the first term is $ a$ and the common ratio is $C r$ : ${S_n}=\dfrac{ a(1-C r^{ n})}{1-C r}$ We are given the value for $ n$. We are also given the first terms in the series, which tells us the values for, $ a$ and $C r$. After we find them, let's plug them in the formula. We are given that ${n=8}$. Since the series is $2+8+32+...$, we can tell that ${a=2}$, and $C{r=4}$. ${S_n}=\dfrac{ 2(1-C 4^{{8}})}{1-C 4}$ Evaluating the expression in the calculator, we get that $S_n=43{,}690$. In conclusion, the sum of the first $8$ terms in the series is $43{,}690$.